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Physical Review A

Expansion of scattering length in S -matrix poles and the phenomenon of resistant virtual states

Roman Čurı́k, Michal Tarana, J. Horáček

Published Jul 11, 2023• 5 citations
Publisher

Abstract

A simple expansion of the scattering length in terms of $S$-matrix poles is derived for the angular momenta $l>0$. This expression shows that the dominant role in the low-energy collisions is played by the poles lying close to the origin of the complex momentum plane. Among these poles a peculiar class of virtual states is found to respond very weakly to the scaled perturbation potentials and thus resistant to producing trajectories in the complex momentum plane. Properties and an impact on low-energy collisions of these resistant virtual states are discussed.

DOI
10.1103/physreva.108.012807

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A simple expansion of the scattering length in terms of SSS-matrix poles is derived for the angular momenta l>0l>0l>0. This expression shows that the dominant role in the low-energy collisions is played by the poles lying close to the origin of the complex momentum plane. Among these poles a peculiar class of virtual states is found to respond very weakly to the scaled perturbation potentials and thus resistant to producing trajectories in the complex momentum plane. Properties and an impact on low-energy collisions of these resistant virtual states are discussed.
The numerical approach to calculating the SSS-matrix poles associated with virtual states in electron-molecule collisions is proposed within the modified effective-range theory frame. The SSS-matrix continuation into the complex momentum plane is possible thanks to the analytical properties of modified Mathieu functions, that is, the exact solutions of the Schr\"odinger equation with the long-range polarization potential ($\ensuremath{\sim}{r}^{\ensuremath{-}4}$). The short-range electron-molecule interaction is included in the introduced numerical approach by model comparison with experimental integral, differential, and momentum-transfer elastic cross sections. The influence of the polarization potential on the virtual-state pole positions is analyzed for nonpolar targets such as N2,CH4,CO2{\mathrm{N}}_{2}, {\mathrm{CH}}_{4}, {\mathrm{CO}}_{2}N2​,CH4​,CO2​, and SF6{\mathrm{SF}}_{6}SF6​. The relation between the SSS-matrix pole position and the sss-wave scattering length is discussed.
Abstract Modified effective range theory is applied as a tool to determine bound and virtual state energies in low-energy positron elastic scattering by atoms and molecules. This is achieved by the S-matrix continuation into the complex momentum plane, allowing to identify poles related to shallow energy states. The influence of the long-range polarization potential (∼r −4) on the bound and virtual-state pole positions is analyzed for noble gases and nonpolar molecules such as H2, N2, and CH4. The quantitative relations between the S-matrix poles and the s-wave scattering length accounting for dipole polarization are introduced.
The complex eigenvalues of some non-Hermitian Hamiltonians, e.g., parity-time-symmetric Hamiltonians, come in complex-conjugate pairs. We show that for non-Hermitian scattering Hamiltonians (of a structureless particle in one dimension) possessing one of four certain symmetries, the poles of the SSS-matrix eigenvalues in the complex momentum plane are symmetric about the imaginary axis, i.e., they are complex-conjugate pairs on the complex-energy plane. This applies even to states which are not bounded eigenstates of the system, i.e., antibound or virtual states, resonances, and antiresonances. The four Hamiltonian symmetries are formulated as the commutation of the Hamiltonian with specific antilinear operators. Example potentials with such symmetries are constructed and their pole structures and scattering properties are calculated.
The Efimov phenomenon manifests itself as an emergent discrete scaling symmetry in the quantum three-body problem. In the unitarity limit, it leads to an infinite tower of three-body bound states with energies forming a geometric sequence. In this work, we study the evolution of these so-called Efimov states using relativistic scattering theory. We identify them as poles of the three-particle SSS matrix in the complex energy plane, and we study how they transform from virtual states through bound states to resonances when we change the interaction strength. We dial the scattering parameters toward the unitarity limit and observe the emergence of the universal scaling of energies and couplings---a behavior known from the nonrelativistic case. We additionally find that Efimov resonances follow unusual, cyclic trajectories accumulating at the three-body threshold and then disappear at some values of the two-body scattering length. We propose a partial resolution to this ``missing states'' problem.
Perturbation expansions of the response function S(q,\ensuremath{\omega}) of a many-body system in orders of any chosen component of the Hamiltonian are obtained. The response S(q,\ensuremath{\omega}) describes the inclusive scattering of a projectile interacting weakly with a target, where q and \ensuremath{\omega} are the momentum and energy transfer. Each term of a given order is written as an expectation value of products of paired Green's functions representing each initial- and final-state effects. For several choices of the perturbing operator, close expressions are obtained for the first orders of these expansions. The resulting terms represent partial summations over the Gersch series, whose relation to Green's-function expansions is established. Also, they are finite for singular potentials and are useful in the asymptotic regime of large q. In this regime they can give the dominant contribution to the response and initial- and final-state interactions of the hit particle and other particles in the system may be fully accounted for. Contributions of the n-particle density matrix can be separated for some expansions. In particular, an expression for the complete two-body contribution involving the hit particle is given.